Existence of resolvable path designs (Q923105)

Let \(\lambda K_ n\) denote the complete multigraph on n vertices in which every edge has multiplicity \(\lambda\), and let \(P_ k\) denote the path with k vertices. A \(P_ k\)-factor of \(\lambda K_ n\) is a spanning subgraph each component of which is isomorphic to \(P_ k\). The authors prove that the edge set of \(\lambda K_ n\) has a partition into \(P_ k\)-factors, \(k\geq 2\), if and only if \(n\equiv 0(mod k)\) and \(\lambda\) k(n-1)\(\equiv 0(mod 2k-2)\).